Biomedical Engineering Reference
In-Depth Information
flexible two-dimensional axisymmetric spheres, the principal bending moments can be
defined as
G
ð
K
1
1
ν
K
2
Þ
M
1
5
λ
2
ð
6
:
20
Þ
G
ð
K
2
1
ν
K
1
Þ
M
2
5
λ
1
where G is the bending stiffness,
λ
n
is the extension ratios, and K
n
are
the changes in curvatures of the membranes. The relationship between the stress and the
extension ratios are also needed for these solutions. These are typically defined as
ν
is Poisson's ratio,
3
1
E
ðλ
2
λ
Þ
1
T
1
5
T
O
1
2
λ
2
ð
6
:
21
Þ
3
2
E
ðλ
2
λ
2
Þ
T
2
T
O
5
1
2
λ
1
for the principal directions, where E is the elastic modulus and T
O
is an initial stress. Once
the cell membrane mechanical properties are defined, we need to classify the properties
for the interior of the cell. Typically, the interior is assumed to be a uniform Newtonian
fluid; that way, forces that act on the cell can be transmitted evenly throughout the cell
interior. It is also assumed that the total surface area of each cell remains constant during
any deformation.
Using these assumptions many groups have studied how red blood cells deform
through the microvasculature at different viscosities, pressure differences, blood vessel
diameters, and hematrocrits. In general, they have all found similar results which suggest
that the pressure within the blood vessel depends very strongly on the cell diameter. With
greater cellular packing, the pressure across the vessel increases as well. They have also
verified that the cells typically stream toward the centerline and that they have a higher
mean flow than the fluid.
The second critical parameter to consider while modeling cell-wall interactions is the
adhesion probability between the blood cell and the vessel wall. Adhesion between cells
and the blood vessel wall is mediated through cell membrane bound proteins, usually
selectins. There is no easy way to quantify the amount of selectins expressed on the cell
membrane and the location of the protein on the cell membrane. To account for this prob-
lem, most computational models use probability theory to help predict where and when
the molecules will be expressed. If the quantity of adhesion molecules is not held constant
in the model, then there will be some criterion for expression; for instance, when the shear
stress level exceeds 40 dynes/cm
2
, x many adhesion molecules can be expressed. Once
this criterion is met, then the model will have a probability statement which states that
some percentage of cells will express the adhesion protein at a particular rate. If the crite-
rion for expression persists, then more cells will continually express more of the adhesion
molecule. However, if the criterion is transient, then the majority of cells will not express
the adhesion molecule, and there are likely statements for the probability of an expressed
protein to be recycled within the cell. After the adhesion molecules have been expressed,
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